In general, carbon fibers are industrially produced by using rayon, PAN (polyacrylonitrile) or a pitch as a raw material. PAN is disadvantageous as it is expensive and the carbonization yield is low. In this respect, pitches are inexpensive and thus economically attractive. However, low cost carbon fibers produced from isotropic pitches are poor in orientation and, therefore, low in strength. These pitches cannot provide a high performance fiber. In contrast, carbon fibers produced from optically anisotropic pitches which are calles mesophase pitches have a highly orientated structure as well as excellent mechanical characteristics, i.e., high strength and a high modulus of elasticity. Thus, extensive studies have been made to produce mesophase pitches as raw materials for high performance carbon filbers from heavy oils such as petroleum catalytic cracking residual oil, naphtha tar pitch, or coal tar pitch. It has been confirmed by various experiments that when mesophase pitches are spun, molecules mainly composed of polycondensed aromatics are orientated in the direction of the fiber axis and thus high performance carbon fibers can be obtained. In order to produce high performance carbon fibers from pitches on an industrial scale, the pitches naturally should have a well-developed polycondensed aromatic structure wherein they are highly optically anisotropic. Additionally, from the industrial viewpoint of stable spinning for a long period of time, it is indispensable that pitches should have a sufficiently low spinning temperature in view of their decomposition temperature. In other words, the pitches should have a low softening point. Also, to improve the spinning properties, it is important that they should be homogeneous in quality and desirably in a 100% optically anisotropic phase. However, the mesophase pitch has disadvantages in that the viscosity is high and, therefore, the softening point is high because of the high interaction of polycondensed aromatics. For this reason, various studies have been conducted to improve the spinning properties of the mesophase pitch by lowering its softening point. When, however, a complicated mixture such as a petroleum pitch or a coal tar pitch was used singly as a starting material and subjected to a thermal modification for the purpose of developing a polycondensed aromatic structure, the resulting pitch did not have a significantly low softening point. In many cases, the mesophase pitch has been spun at temperatures as high as 340.degree. C. to 380.degree. C. At such high temperatures, however, the pitch is liable to undergo thermal decomposition and a thermal condensation reaction during the spinning process, thereby producing gas and high molecular weight substances. Thus, it has been difficult to carry out stable spinning of the mesophase pitch for long periods of time. In some instances, an attempt has been made to lower the softening point of the pitch at the sacrifice of its optical anisotropy.